Group Size: Any

Learning Objectives:

1. Students will develop an appreciation for the applicability of systems of linear equations to real-world problems.

2. Students will use various methods of solving systems of equations to determine break-even points for proposed product prices in their project.

Materials:

Final drafts of cost curves and linear regressions from student portfolios

Graph paper

Rulers

Colored pencils

Digital projector connected to a computer with Microsoft Powerpoint

Attached Powerpoint presentation

Procedures:

Inform students that today is the day that we will be putting all of the various Algebraic methods together to make meaningful predictions and decisions about our project. Have them remind you what the cost curves represent. Next, remind them about revenue curves. Elicit the fact that the equation for a revenue curve is simply:

y = (Price) x

where x stands for the # of units sold and y represents \$

Thus, for a given price, a revenue curve is an extremely easy equation to graph. (e.g. y = 1.25x for a price of \$1.25 per unit).
Remind students what we addressed when first learning about systems of equations: break-even points.
Lead a brief discussion to remember and understand the importance of finding the break-even point for their product. (Remind them that they will be making a proposal for a loan to the principal, and he will primarily be interested in how many units they will need to sell in order to pay back the loan.)
Highlight that the break-even point will be different for each potential price, because the amount of money made will be higher for a given number of units with a higher price. Thus, in order to select an appropriate price for our goods, we will need to analyze several different potential prices and determine which price we feel we are most likely to be able to sustain in our market. (Remember, if we set a lower price, we will probably sell more, but we will need to sell more in order to break even. Conversely, if we set a higher price, we will make more revenue, but we will most likely sell less because of the nature of demand! Thus, picking the right price is a balance between making marginal revenue (the revenue per unit sold) and not over-taxing the demand!)
Explain to the class that they are going to need to prepare a presentation for the school principal to convince him/her to give us a loan for the start-up costs. In order to convince them, we will need to have a convincing proposition, know exactly how much money we are asking for (that is, the amount of money represented by the break-even point for the price we select), have a clear business plan (that is, how are we actually going to run the business), and have a convincing presentation. We are going to work on preparing the business plan and the presentation over the next several class periods.
Use the attached powerpoints to give students an idea of what their presentation could look like. Also, pass out the attached business plan to provide the class with a model of what the business plan should look like. Go through the powerpoint and present it as if you were presenting to a loan officer. Afterwards, go through slide by slide and explain the method for acquiring the data.
Explain to the students that they will learn how to generate the cost/revenue curve graphs in a later lesson. For now, they need to be able to create these graphs on paper.
After giving them an overview of the presentation and business plan, facilitate a class discussion on the essential elements of their business plan. After these have been made clear, have each group write up a business plan collaboratively and present it. The "best" business plan (according to a vote by the class) will be the one presented to the principal.
Next, have each student get graph paper and colored pencils. Write the data from the grilled cheese proposal on the board:
Cost Curve Equation : y = 20 + 0.57x
Revenue Curves:
Selling sandwiches at \$1 apiece: y = 1x
Selling sandwiches at \$1.25 apiece: y = 1.25x
Selling sandwiches at \$1.50 apiece: y = 1.5x
Selling sandwiches at \$1.75 apiece: y = 1.75x
Selling sandwiches at \$2 apiece: y = 2x
Now, have students graph these 6 equations on their graph paper. (Lead a discussion on the appropriate scale for the graph.)
Have students use a different color for each revenue curve, so that they can easily refer to the different prices and break-even points in their analysis.
After they have generated a pencil-and-paper system of 6 equations, have them find the points of intersection between the cost curve and each of the 5 revenue curves using an Algebraic method of solving (either substitution or linear combination). Tell them to record the Algebra neatly on the same sheet of graph paper (perhaps on the back) and to explain what it all means. Their work should be so clearly explained that someone who is unfamiliar with the project should be able to understand what the graphs and equations stand for and what the solutions represent.
Once they have completed this, have them show you their work. If it is satisfactory, have them create the same graph and Algebraic analysis, using the cost curve and potential price/revenue curves the class has generated. Their final product should be a clear, easily understood analysis of the costs and potential revenues associated with selling their product, as well as a clear delineation of where the break even points (both the number of units needing to be sold and the amount of money required in costs) for each different price are.
When they complete this, they should put in their portfolio to be graded at the end of the project.

Attached Files: